Why do two likes give a positive, and two unlikes give a negative

• Oct 27th 2010, 01:33 AM
Kick
Why do two likes give a positive, and two unlikes give a negative
The rule i learned is:

For multiplication or division two likes give a positive answer, two unlikes give a negative answer.

E.g.
-2 * -2 = 4

-2 * 2 = -4

BUT look at this...

2 * 2 = 4 is the same as saying 2 + 2 = 4. Right?So....

-2 * -2 = 4 should be the same as saying -2 + -2 = x

But... -2 + -2 = -2 - 2 = -4 ..... so it isn't the same pattern as above.

Can anyone shed some light?
• Oct 27th 2010, 02:11 AM
Also sprach Zarathustra
Quote:

Originally Posted by Kick
The rule i learned is:

For multiplication or division two likes give a positive answer, two unlikes give a negative answer.

E.g.
-2 * -2 = 4

-2 * 2 = -4

BUT look at this...

2 * 2 = 4 is the same as saying 2 + 2 = 4. Right?So....

-2 * -2 = 4 should be the same as saying -2 + -2 = x

But... -2 + -2 = -2 - 2 = -4 ..... so it isn't the same pattern as above.

Can anyone shed some light?

A mistake here:

Quote:

-2 * -2 = 4 should be the same as saying -2 + -2 = x
Think more...
• Oct 27th 2010, 04:23 AM
HallsofIvy
What kind of answer do you want? You can show from the "field axioms" that the product of two negative numbers is positive. That's very simple, completely correct and very technical- requiring that you know a lot of definitions and abstract concepts you probably don't.

Or you can argue that, just as if you have, say, 5 boxes, each containing 10 dollars, then you have 5*10= 50 dollars, that if you owe \$10 to each of 5 people (so - \$10), and each of the 5 people cancels ("negates") the debt, you no longer owe any thing- its just as if you had suddenly gained \$50 to pay off your debts: (-\$10)(-5)= +\$50. That's very basic, very simple, and very "hand-waving".

But that's the way mathematics is- you can have technical and precise or non-technical and vague.
• Oct 27th 2010, 04:03 PM
Kick
Quote:

Originally Posted by Also sprach Zarathustra
A mistake here:

Think more...

I have thought. Where is the mistake.

If 2 * 2 = 4 is the same as saying 2 + 2 = 4 then...

-2 * -2 = 4 is the same as saying -2 + -2 = 4. But the last equation does not equal 4, it equals -4. -2 + -2 = -4.

So what is going on?
• Oct 28th 2010, 12:39 AM
Pim
The flaw is in that the rule a*b = a+a+a+a+... (and that b times) can't be a applied to negative numbers.
You can't say: I'll add -2 to itself -2 times. You can't do something a negative amount of times.

How HallsofIvy explained it is a better way to think about it.
• Oct 28th 2010, 02:32 AM
brennan
Imagine Do is +, Don't is -
Imagine Win is +, lose is -
Imagine we a gambling - so we either win or lose.

2 times I do(+) win(+) £10 : so I win(+) £20 : so +x+ = +

2 times I dont(-) win(+) £10: so I lose(-) £20: so -x+ = -

2 times I do(+) lose(-) £10 : so I lose(-) £20: so +x- = -

Here is you tricky bit.

2 times I dont(-) lose(-) £10 : so I win(+) £20: so -x- = +
• Oct 28th 2010, 02:49 AM
Quote:

Originally Posted by Kick
I have thought. Where is the mistake.

If 2 * 2 = 4 is the same as saying 2 + 2 = 4 then...

-2 * -2 = 4 is the same as saying -2 + -2 = 4. But the last equation does not equal 4, it equals -4. -2 + -2 = -4.

So what is going on?

You are using the * operator as an addition operation.

"2 times 2" maybe ordinarily worded as 2 "two times with a plus between".

2(2) or 2*2 is 2+2.
3(2) or 3*2 is 2+2+2 or 3+3
4(2) or 4*2 is 2+2+2+2 or 4+4

-2*2 is a symbolic way to represent a "double subtraction of 2"
-2-2=2(-2)=2(-1)2=(-1)4
Basically it is "subtract 2 twice" in common language (just one way to describe it).

-2*3=-2-2-2
-2*4=-2-2-2-2

(-2)*(-2)=-[2(-2)]=-(-2-2)=-(-4)

"-" is the opposite of, if you'd like to think of it that way.

So you have the "opposite of subtract 2 twice" which is....
• Oct 28th 2010, 08:40 AM
wonderboy1953
Quote:

Originally Posted by Kick
The rule i learned is:

For multiplication or division two likes give a positive answer, two unlikes give a negative answer.

E.g.
-2 * -2 = 4

-2 * 2 = -4

BUT look at this...

2 * 2 = 4 is the same as saying 2 + 2 = 4. Right?So....

-2 * -2 = 4 should be the same as saying -2 + -2 = x

But... -2 + -2 = -2 - 2 = -4 ..... so it isn't the same pattern as above.

Can anyone shed some light?

What you're really asking is why is the product of a negative and a positive number defined to be a negative number and why is the product of two negative numbers defined to be a positive number?

Let's do a pattern:

4 x 4 = 16
4 x 3 = 16 - 4 = 12
4 x 2 = 12 - 4 = 8
4 x 1 = 8 - 4 = 4
4 x 0 = 4 - 4 = 0
4 x -1 = 0 - 4 = -4...

let's do another pattern:

-4 x 4 = -16
-4 x 3 = -16 + 4 = -12
-4 x 2 = -12 + 4 = -8
-4 x 1 = -8 + 4 = -4
-4 x 0 = -4 + 4 = 0
-4 x -1 = 0 + 4 = 4...

If you see the patterns and you know a little algebra to generalize the results, then you have your answer.